Scowcroft opened the scoring when he picked up Dion Dublin's flick-on after eight minutes before Dabizas headed home 21 minutes later.
Wilcox pounced on a loose ball in the second half to make it 3-0 before Chris Morgan grabbed one back for the Blades.
Jonathan Forte lashed the ball home in the 84th minute but Leicester survived.
Sheffield United boss Neil Warnock:
"We committed football suicide, we gave away three goals
thanks to schoolboy defending.
"You go anywhere in this league and leak goals like that then you have no
right to come away with anything.
"But then we showed our character to come back from that and in the end we
were disappointed not to have got something out of the game."
Leicester: Pressman, Makin, Dabizas, Keown, Stewart, Scowcroft, Nalis, Williams, Wilcox (Tiatto 88), Dublin, Connolly (Blake 88).
Subs Not Used: Wright, Gemmill, Benjamin.
Goals: Scowcroft 8, Dabizas 29, Wilcox 52.
Sheff Utd: Kenny, Bromby, Jagielka, Morgan, Wright (Forte 46), Quinn, Montgomery, Thirlwell, Harley, Gray (Lester 64), Ward.
Subs Not Used: Barnes, McCall, Hurst.
Goals: Morgan 75, Forte 84.
Ref: P Crossley (Kent).